模4整数上编码的形式幺模格的保密增益
Secrecy Gain of Formally Unimodular Lattices from Codes over the Integers Modulo 4
AI总结：
针对Gaussian wiretap channel中Construction A4格的保密增益问题，提出了偶数和奇数长度下形式自对偶Z4-线性码的新构造，推导了形式幺模格的theta级数并给出确定保密增益的通用方法，证明其能实现比已知最佳格更高的保密增益。
摘要：
最近，一种依赖于格的体积和theta级数的设计准则被提出，称为保密增益，用于量化应用于Gaussian wiretap channel的格码的保密优良性。为了解决由形式自对偶Z4-线性码（即对称化重量枚举子（swe）与其对偶码的swe相同的码）构造的Construction A4格的保密增益问题，我们提出了在swe方面形式自偶的Z4-线性码的新构造。对于偶数长度，形式自对偶Z4-线性码由嵌套二进制码和双循环矩阵构造。对于奇数长度，提出了一种称为双循环码奇扩展的新颖构造。此外，引入了Type I/II形式自对偶码/幺模格的概念。接下来，我们推导了由形式自对偶Z4-线性码通过Construction A4获得的形式幺模格的theta级数，并描述了确定其保密增益的通用方法。我们研究了偶数和奇数维度下，由形式自对偶Z4-线性码获得的Construction A4形式幺模格的保密增益。数值证据表明，对于某些参数，Construction A4格能够实现比文献中已知最佳形式幺模格更高的保密增益。还讨论了与flatness factor（Gaussian wiretap channel中广泛考虑的另一个安全准则）相关的结果。
英文摘要：
Recently, a design criterion depending on a lattice's volume and theta series, called the secrecy gain, was proposed to quantify the secrecy-goodness of the applied lattice code for the Gaussian wiretap channel. To address the secrecy gain of Construction $\text{A}_4$ lattices from formally self-dual $\mathbb{Z}_4$-linear codes, i.e., codes for which the symmetrized weight enumerator (swe) coincides with the swe of its dual, we present new constructions of $\mathbb{Z}_4$-linear codes which are formally self-dual with respect to the swe. For even lengths, formally self-dual $\mathbb{Z}_4$-linear codes are constructed from nested binary codes and double circulant matrices. For odd lengths, a novel construction called odd extension from double circulant codes is proposed. Moreover, the concepts of Type I/II formally self-dual codes/unimodular lattices are introduced. Next, we derive the theta series of the formally unimodular lattices obtained by Construction $\text{A}_4$ from formally self-dual $\mathbb{Z}_4$-linear codes and describe a universal approach to determine their secrecy gains. The secrecy gain of Construction $\text{A}_4$ formally unimodular lattices obtained from formally self-dual $\mathbb{Z}_4$-linear codes is investigated, both for even and odd dimensions. Numerical evidence shows that for some parameters, Construction $\text{A}_4$ lattices can achieve a higher secrecy gain than the best-known formally unimodular lattices from the literature. Results concerning the flatness factor, another security criterion widely considered in the Gaussian wiretap channel, are also discussed.
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